Montessori Basics: Geometry from the Start

“We do not therefore offer material for the clear and concrete demonstration of what is taught in a an abstract fashion in most schools. We simply offer geometric shapes, in the form of material objects, which have a relationship to each other. These shapes can be moved and handled, lending themselves to demonstrating or revealing evident correspondences when they are brought together and compared. This stimulates mental activity, because the eye sees and the mind perceives things that a teacher does not know how to convey to an immature and inactive mind. Mental processes that are apparently premature and far advanced for the child’s age, thus become possible.” - Maria Montessori Excerpt from Psychogeometry, Chapter on “Introduction to the Elementary Period,”

Perhaps it happens when your four-year-old comes home from school one day, excited to show you their work for the day.  They proudly show you a perfectly traced pentagon with elaborate, colorful patterns inside that they have created.

Maybe it’s when your eight-year-old casually references acute-angled scalene triangles.

Regardless of when it happens, as Montessori parents, there comes a moment when we become acutely aware (pun intended) of our children’s interesting knowledge of geometry.  We may recall our own study of the subject beginning much later - likely sometime during our high school years and typically not as exciting as our own children depict! We notice that our children seem to be really ready for the information, which can feel surprising.  Not only are they ready, but the work seems to fill them with joy and satisfaction.

What, exactly, is going on?

As with so many things, Montessori discovered that young children are fully capable, and in fact developmentally primed, to learn about subjects that have traditionally been reserved for much older children.  Geometry is a perfect example. Read on to discover what this portion of a Montessori education can offer your child.

The Primary Years

From ages 3-6 much of children’s geometry instruction in Montessori classrooms is indirect.  That is to say that while they are practicing crucial developmental skills, they are often doing so through the lens of geometry preparation.  One obvious example, as mentioned above, is with the metal insets. Children trace a variety of geometric figures including squares, triangles, circles, curvilinear triangles, and quatrefoils, among others.  The main objective of this work is to prepare the child’s muscles for proper pencil grasp and handwriting. When they have mastered tracing they work to create intricate designs within the figure.  

Primary children are also given a number of simple geometry lessons that allow them to begin naming figures and exploring shapes.  Wooden geometric solids are held and named by the children (cube, sphere, square-based pyramid, etc.). The geometry cabinet is composed of drawers of related figures; small wooden insets are organized into a polygon drawer, curvilinear figure drawer, triangle drawer, and so on.  Children also use constructive triangle boxes to manipulate triangles in order to form larger triangles and other geometric figures. The key during these early years is to give children early exposure to geometry and allow them to use their hands to explore these concepts.

The Elementary Years

During the elementary years the Montessori geometry curriculum expands significantly.  Teachers often begin by reviewing content taught during the primary years, but 6-year-olds are ready and eager for more.  This begins with a detailed study of nomenclature. Using a series of cards and booklets that correspond with lessons given by the teacher, children explore and create their own nomenclature sets.  Topics include basic concepts such as point, line, surface, and solid, but go on to teach more in-depth studies of lines, angles, plane figures, triangles, quadrilaterals, regular polygons, and circles.  For example, when children learn about lines they begin by differentiating between straight and curved lines, but go on to learn concepts such as rays and line segments, positions (horizontal and vertical), relational positions of lines (parallel, divergent, perpendicular, etc.)

Throughout the second plane of development (ages 6-12) the study of geometry continues to spiral and go into more and more depth.  Children as young as seven learn about types of angles and how to measure them. Eight-year-olds explore regular and irregular polygons, as well as congruency, similarity, and equivalency.  In lower elementary children begin learning about perimeter, area, and volume.

In upper elementary, children begin to learn about the connections between the visual aspects of geometry and numerical expressions.  They apply what they’ve learned about perimeter, area, and volume to measuring real-life objects - including Montessori materials they’ve seen in their classrooms since they were three years old.  They learn about things like Fibonacci numbers and Pythagoras which appeal to their sense of number order and geometric patterns.

Now, when your child comes home with surprising knowledge about geometry content, we hope you have a better idea of where they’re coming from.  If you have any questions or would like to see this type of work in action, please give us a call.

Observation

Montessori classrooms rely heavily on the art of observation.  You may see it in action some time, or you may have an opportunity to try it yourself (which we welcome and encourage!).  If you ever walk past a classroom and see the children working intently, while the guide is quietly sitting in a corner with a clipboard, know that guide is working intently as well.  

Why we do it

Dr. Maria Montessori was a scientist and a physician.  Her education and background helped her look at the world in a way that is different from most traditional educators.  Observation of children was what inspired her work in education, and she used it to develop her methods. Not only that, but Montessori guides all over the world rely on observation to learn about their students, gain insight about developmental phases, inform our decision-making, and to assess the children’s mastery of skills.  So what are the main goals of observation in the classroom?

• Planning appropriate lessons - Montessori educators are trained to have extensive knowledge about child development.  While most traditional teacher education programs require students to take a course on the topic, development is essentially the foundation of everything we do as Montessori guides.  Practiced guides know so much about the behaviors of growing children that seemingly insignificant occurrences signal a transition into a new plane of development. The toddler that has mastered toileting and can be observed spending long amounts of time with practical life activities is making the transition necessary for the primary classroom.  A child nearing six that has lost a tooth and seems suddenly very motivated by social interactions with their peers is moving into the second plane of development and will respond well to lessons involving storytelling and deeper information about cultures around the world. The challenge of the guide is to identify the moment when a child is entering a sensitive period; this is their development showing they are ready to learn specific skills that must be taught in a way that honors their growth.  

• Making sure the environment serves the children - While the guide’s role in a child’s education is important, the environment plays an even bigger role.  It is the guide’s job to make sure the classroom environment allows children to find what they need, feel inspired to work, fosters independence, and allows for safety and comfort.  If, during an observation, it is noted that a piece of furniture disrupts the flow of movement, it will be moved at a later time. If many children prefer to work on the floor, it will be important to note whether there are enough work rugs for them all to use.  If no children have used a specific material in a number of weeks, it may need to be reintroduced or removed from the shelf. Each item in the classroom must be placed intentionally and with a specific purpose in mind. If it is no longer serving its intended purpose, reflection and a solution are required.

• Assessment of skills - The word assess is derived from the Latin form to sit beside.  Montessori schools do not determine mastery with the use of tests, but rather by utilizing observation.  Instead of giving children a piece of paper with questions on it, we watch them in action. When a child is able to independently place number tiles in random order on a hundred board, we know they have grasped the concept of ordering those numbers.  A child who is able to complete complex patterns within the shape they traced using a metal inset, and who also frequently uses the sandpaper letters correctly is likely ready to learn the written formation of letters using a pencil on a piece of paper.  This assessment, of course, ties back into planning appropriate lessons, as the guide has concrete information to inform their instruction.

What it looks like

• Formal observations - A Montessori guide will likely observe in the classroom most days, or multiple guides may take turns observing.  These observations typically last between fifteen and thirty minutes, but the amount of time can vary. While each guide has their own preferred method, they typically sit quietly and use a notebook to record what they observe.  Children are taught about the importance of this work and they know not to disturb the adult at this time. Sometimes a guide will sit in a specific chair or use a special clipboard to signal to the children that they are working.  For new guides, the temptation to intervene can be powerful, but we learn that unless a child is in danger it’s often best to wait it out and see what happens. Most classrooms have a second adult that is able to redirect a child who may be overly disruptive, allowing the observing adult to continue.  During this time the guide simply watches and takes lots of notes. It is important that the notes be strictly observational and that any judgement or inferencing be reserved for another time.

• Informal observations - During the course of the work period, guides will make a great many observations in the moment.  While walking across the room to retrieve something, while speaking to a child, or even whilst in the middle of a lesson, there are many helpful bits of information a guide can gather and record that will help make the children’s educational experience the best it can be.  As you may imagine, this results in many, many notebooks full of amazing and adorable anecdotes. 

Visitor observation

Whether you are considering Montessori for your child, they are in a program but getting ready to move to a new level, or if you’re just curious and want to learn more about the philosophy, observation is one of the greatest tools available to you.  Even the most experienced guides make time to visit other Montessori schools when possible so that they may observe other classrooms and gather fresh ideas and inspiration. We invite you to contact us should you be interested in giving it a try.

When you enter a Montessori classroom to observe, it is very important to know that the children will be engaged in their work and the goal is to watch without disturbing them.  In many other scenarios in life, we announce ourselves upon entering a room, perhaps even greeting others enthusiastically. When observing in the classroom, we ask that visitors refrain from doing these things, tempting as it may be!  You will likely be greeted by an adult or child and directed to a chair. Having a notebook or clipboard is helpful, as you are sure to experience moments you will want to record. If a child approaches you and greets you, by all means please feel free to briefly greet them in return.  In general, however, you will need to sit quietly and observe in a way that the children forget you are there, leaving them free to focus on their work. Montessori children are quite used to visitors, so this shouldn’t be a problem.

Some questions that will help guide your observation include:

• In what ways are the children displaying independence?

• How do the children choose their work?

• What do transitions between work look like?

• How do the adults respond to the children?

• How do the children respond to the adults?

• How do the children interact with one another?

• How do the children care for their own basic needs?

• Does anything about the classroom environment surprise you?

When your observation is complete, it is best to slip out of the classroom quietly.  In this situation you are not expected to say any formal goodbyes.

You can do this at home!

While home is very different from the classroom, there are ways that parents can apply the basic concepts of Montessori observation.  While trying to engage with our children, it can be easy to fall into patterns in which we begin directing their play. Every once in a while, sit back and watch as your child plays.  You may notice them using their toys in surprising ways, and this may give you insight to their interests and maturity. Similarly, it can be tempting to jump in and help any time your child spills something, falls down, or struggles to do something.  Instead of rushing to the rescue, wait. If they ask for help, of course, lend a hand, but oftentimes they will want to address the situation themselves. Watching to see what our children are capable of and nurturing their independence is one of the greatest gifts you can give them.  

As you pay attention and observe your child’s play, eating habits, sleeping habits, and social habits, you may learn many new things about their development.  This, in turn, will allow you to reflect on how you might best support them on their journey through childhood. Slow down, observe, and enjoy those moments.

5 Ways Montessori Appeals to the Senses

“The senses, being explorers of the world, open the way to knowledge. Our apparatus for educating the senses offers the child a key to guide his explorations of the world…” The Absorbent Mind p. 167

Learning with all our senses involved allows us to have a fuller, richer experience.  Montessori classrooms strive to provide multi-layered sensory opportunities for children.  The result? Children who have a strong ability to distinguish the variances in the environments around them.

1. Montessori digs deeper than the classic five senses.

Growing up, you undoubtedly learned about sight, hearing, taste, touch, and smell.  Of course, these are the five basic senses we tend to think about, but Montessori education has a more extensively defined list all its own: 

• Visual - our ability to differentiate objects by form, color, and size

• Tactile - just another name for the sense of touch, or how something feels on our body

• Baric - differentiation based on weight and/or pressure

• Thermic - the ability to sense various temperatures

• Auditory - another name to describe the sense of sound

• Olfactory - our sense of smell

• Gustatory - the sense of taste

• Stereognostic - a muscular sense, or the ability to distinguish an object without seeing it, hearing it, or smelling it, but relying of touch and muscle memory alone

2. Montessori developed materials to help children refine their senses.

Using what she knew about the above senses, Dr. Montessori developed a series of sensorial materials to be used in the classrooms of young children.  These materials were designed to isolate one skill and to be self-correcting. This allows the child to concentrate their efforts and to be independent in their learning.   Just a small selection of the more famous sensorial materials include: 

• Knobbed Cylinders - small wooden cylinders with knobs that are to be inserted into holes of the corresponding size

• Pink Tower - a series of pink wooden cubes ranging in size from 10 cm cubed to 1 cm cubed are meant to be stacked in decreasing succession

• Brown Stair - ten brown, wooden rectangular prisms in a range of sizes are meant to be arranged in order

• Color Tablets - a material that allows children to differentiate not just by color, but by shades of colors

• Mystery Bag - children are meant to reach their hand inside the bag without looking to determine the contents

• Geometric Solids - a physical representation of an often abstractly-taught concept, these solids allow children to identify their attributes

3. Food is prepared and celebrated regularly in Montessori classrooms.

Beginning when they are just toddlers, Montessori children are directly involved in the preparation and purposeful enjoyment of food.  Toddler classrooms have regular tastings, in which they try new and interesting foods. Guides will offer a wide variety of textures, colors, smells, and tastes for the children to explore.  These little ones help set the table and learn grace and courtesy through table manners.

During the primary grades (ages 3-5), children participate in food preparation.  They are given lessons and chances to practice slicing, spreading, mixing, blending, and multi-step food preparation.  Sometimes they enjoy their work as a snack for themselves; other times they prepare food to serve to others.

Guides in older levels find ways to continue this important work.  Food preparation may be connected to a cultural study, birthday celebration, or school lunch program.  As they get older, children are able to complete more complex and interesting recipes. 

4. The classroom environment keeps a focus on the natural world.

Montessori guides are taught to make nature an integral part of the classroom environment, and this often means lots of beautiful indoor plants.  Studies have shown that proximity to plants benefits us a variety of ways. They are visually beautiful, but did you know that scientists believe that houseplants can improve our attention?  They may also be helpful in reducing sick days and keeping us more productive overall. *See links at the end for more information. 

Aside from having live plants in our classrooms, Montessori schools favor natural materials over synthetic.  This means that whenever possible, we choose wood, glass, and natural baskets over plastic. We believe that the color and texture of natural materials is more appealing and calming to our senses.  While many conventional classrooms favor bright colors, we opt for more muted, natural ones. This allows children to feel calm, safe, and able to focus on their work.

Whenever possible, Montessori schools believe in the importance of taking children into nature on a regular basis.  Whether to a local pond, for a walk in the woods, or even a nearby city park, being in green spaces is an important part of learning and growing. 

5. Montessori honors children’s developing vestibular and proprioceptive systems.

A couple quick definitions- 

The vestibular system is responsible for balance and is closely connected to the inner ear. 

The proprioceptive system is important when having awareness of where one’s body parts are in relation to the rest of one’s body and the space/objects around it.

These systems typically develop early in childhood.  It’s our job as adults to make sure children have opportunities to refine them.  It is especially important that we provide opportunities to children with sensory related disorders.

Although many schools around the country are decreasing or doing away with recess altogether, Montessori schools hold that time in high regard.  All the climbing, swinging, spinning, and other types of play are natural ways for children to develop their vestibular and proprioceptive systems.

There are activities built into Montessori classrooms that assist this work as well.  Carrying heavier materials, painting, and using playdough are connected to the proprioceptive system.  The traditional ‘walking the line’ in Montessori primary classrooms provides excellent vestibular input; children must slowly walk while staying on a taped or painted line.  Extensions include walking with a bell in hand and trying not to ring it or balancing something on top of their head.

Interested in seeing the sensory classroom in action?  Whether you are a current or prospective parent, we encourage you to give us a call and set up a time to observe.  

Sources:

Benefits of Indoor Plants…

https://www.sciencedirect.com/science/article/abs/pii/S0272494410001027?via%3Dihub

Psychological Benefits of Indoor Plants…

http://hortsci.ashspublications.org/content/42/3/581.full